I use a 357 with hottish loads using Hornady 158gr HP-XTP bullets. I figured I could get out to 150 yards with enough accuracy and energy.
I would think the 44 Magnum probably would fit the bill. With 240gr bullets at 100fps faster than my 357 bullets were going, (using the same length barrel,) there is more energy at 250 yards than I have at 150.
The bullets are going subsonic by 150 yards which could cause accuracy problems. I don't seem to suffer from accuracy loss at extended ranges with the same bc and bullet type.
I see pretty many of them in your price range.
Here is a nice one. Smith & Wesson Model 629
There is a Taurus there that would fit the bill for $550... If you like Taurus.
Then you need a scope mount, rings and scope. I priced those items for my pistol at around $200 but I don't know how brutal a 44 magnum would be on a scope and such, you could need really good accessories.